Optimal Ecosystem Management with Structural Dynamics Rui Pedro Mota1, Tiago Domingos, Environment and Energy Section, DEM, Instituto Superior Técnico Abstract We address the problem of optimal management of a self-organizing ecosystem along ecological succession. A dynamic carrying capacity is interpreted as depicting the dynamics of habitat creation and occupation along ecological succession. The ecosystem may have three growth modes: pure compensation (concave ecosystem regeneration function), depensation (convex-concave regeneration function) and critical depensation (additionally having negative growth rates for low biomass). We analyse the optimal policies for the management of the ecosystem for the three growth modes. Accordingly, we prove the existence of a Skiba points for certain types of ecosystems. Further, we compare usual golden rule paths with the derived optimal policies near the Skiba points. Keywords: Ecosystem management, habitat creation, optimal policies, Skiba point. JEL Classification: Q20, C61, C62 1. Introduction In this work we follow a theoretical approach in the context of ecosystem management to deal with the ecosystem dynamics along ecological succession. Ecosystem succession is the process of species change over time. It includes the development of complex systems from simple biotic and abiotic components. Effective manipulations of our environment to improve fertility, productivity or diversity and an ability to mitigate undesirable conditions all depend on our ability to understand ecological succession. In this paper we bring this subject into the realm of economic analysis. We take the perspective of ecosystem management. An ecosystem consists of so many interacting components that it is impossible ever to be able to separate and examine all these relationships (Kay and Schneider, 1994; Jorgensen, 1992). Scientists, land managers, and others are proposing ecosystem management as the best way to manage natural resources. We must, then, explore how this approach differs from the theoretical assumptions of current natural resource management. For ecosystem management, the ecosystem production function is the most useful feature of an ecological model. The ecosystem production function or ecological productivity is simply the graph of dN/dt versus N, being N the ecosystem biomass. It is a highly aggregated representation of an ecosystem’s dynamics that depends on innumerable biological and spatial details (Roughgarden, 1997). The ecosystem 1 Address: Avenida Rovisco Pais, 1, 1049-001 Lisboa, Portugal. Phone: +351 - 218 419 163, Fax: +351- 218 417 365, E-mail: [email protected] 1 production function can be thought of as indicating the interest or return on biomass products and services per unit time (Roughgarden, 1997). We argue that ecosystem production functions should include the dynamics of the structural organization of ecosystems during ecological succession. In order to derive optimal policies along ecological succession we devise a model of an extractive economy using an ecosystem with a dynamics proposed by Rodrigues et al. (2002) and further analyzed in Mota el al. (2004). These kinds of behavior of optimal policies in the presence of convex-concave ecological regeneration functions have also been seen for lake management problems (e.g., Brock and Starrett, 1999; Mäler et al.2003), for fishing models (Dasgupta and Mäller, 2003), or for multi-species models (Scholes, 2003; Crépin, 2003). Here we have somewhat similar results applied to the problem of managing an ecosystem along ecological succession. 2. Ecosystem dynamics In this section we briefly present the main behaviours of an ecosystem whose structure is changing along ecological succession. The following results and figures have been derived in Mota et al. (2004). We interpret the dynamics of biomass as following the dynamics of ecological succession. Odum (1969) notes that ecological succession drives ecosystems’ biomass along a logistic curve and net production along a U-inverted curve. Following this, we assume that the regeneration function for biomass is given by the logistic curve, dN =-rN()CCN, (1) dt where N is ecosystem biomass, CC is the ecosystem carrying capacity and r is a positive constant. This functional form is not the most common form of the logistic but it is more convenient for our purposes. The important feature of ecological succession depicted by the logistic function is the existence of a “reasonably directional development, which culminates in a stabilized ecosystem climax” (Odum, 1969). Ecosystem carrying capacity is the value of biomass attained at this state. It reflects the ecosystem’s organization that best allocates the available resources. However, as Walker and Moral (2003) argue, ‘the more common held view now is that succession is a process of change that is not always linear and rarely reaches equilibrium’. Disturbance often redirects successional trajectories, leading to the observation that stable endpoints are rarely achieved. Walker and Moral citing Cowles (1901) suggest that succession was a ‘variable approaching a variable rather than a constant’. If a new unit of ecosystem biomass can alter the ecosystem carrying capacity, this may be understood as a structural change, since an alteration of the efficiency of the allocation of resources has occurred. This interpretation relates the ecosystem carrying capacity to structural properties of the system. The activities of organisms can result in significant, consistent (non-random), and directed (predictable) changes in their local environment. All living creatures through their metabolism, their activities and their choices, part create and partly destroy their own niches, on scales ranging from extremely local to the global (Odling- Smee, et al., 2003; Jones and Lawton; 1995). Most plants modify their immediate environment in some way that can impact establishment and growth of both other species and other individuals of the same species. Plants selectively concentrate soil nutrients, transport water from the soil to the atmosphere and add organic matter when they decay. Plants also remove resources and impact their neighbors that need the same, often limiting resources. Animals alter soils by burrowing, feeding, defecating and dying (Walker and Moral, 2003, p.189). To take niche construction and specialization along ecological succession into account, Rodrigues et al. (2002) proposed to consider carrying capacity as a dynamic state variable of the ecosystem. The dynamics proposed is: dCCldN = , (2) dtN+hdt and was adapted from a model devised by Cohen (1995) for human populations. The parameter l is the self-organizing capability of the ecosystem and h is the feedback saturation, since it imposes an upper limit to the intensity of feedbacks between ecosystem biomass and ecosystem carrying capacity (Mota et al., 2004). Moreover, if l is high, it means that biomass has strong feedbacks with carrying capacity and small changes in population may have large effects on niche availability. This way, we are endowing the ecosystem with autocatalytic behaviour, since a rise in ecosystem biomass inflicts a rise in ecosystem carrying capacity, which creates a greater potential for biomass to grow (provided that N < CC). Assuming that the dynamics of the ecosystem is governed by (1) and (2), this implies the following ecosystem production function (Mota et al., 2004): dNæöæöNh+ . (3) =rNç÷CCl +llnç÷-ºNRN() dtbèøèø A simple conclusion drawn from the analogy with the logistic model is that this ecosystem has a carrying capacity given by CCl ++lln((Nhb)/ ) which is constant for l = 0 . That is, for zero self-organizing capability, we have the conventional one-population dynamics. This way, the optimum operating point of the ecosystem is not imposed on the system by exogenous conditions but depends on the historical information “remembered” by the system. Feedbacks on carrying capacity may be positive or negative depending on the ecosystem biomass for a fixed l and h. 3 a) b) Figure 3.1 – a) Bifurcation diagram for the ecosystem dynamics for the case where CCbl ³ . p) represents pure compensation; d) non-critical depensation and c) critical depensation. The line 1 hb= is the asymptote of lcrit . b) Bifurcation diagram for the case where b> CCl . e) represents unconditional extinction (Mota et al., 2004). Depending on the feedbacks on carrying capacity, the ecosystem may have different kinds of behavior. The main conclusions are depicted in bifurcation diagrams in the l-h plane, for fixed values of all the other parameters (Figure 3.1). The line lh= separates pure compensation from depensation behaviours. In pure compensation, for all biomass values, each unit added to the existing stock causes a decrease in the per capita growth rate. In other words we have a concave ecosystem production function. In depensation, for sufficiently low N, each unit added increases the per capita growth rate (convex-concave production function). If an ecosystem has depensation behaviour with the additional property of negative growth rates for some biomass levels , it is said to have critical depensation (Clark, 1976, p. 16). In this case, there is a certain threshold biomass stock level below which the ecosystem will become extinct. We conclude that if feedbacks on carrying capacity are low enough (small l and high h) the ecosystem exhibits pure compensation. On the other hand, if feedbacks on carrying capacity are sufficiently high the ecosystem exhibits depensation and,